Jackplug. Known to be annoying after a while when it start getting bad connections.

Wire Cutters or Scissors. Don't run with them.
Cut off as close to the jack as possible if you don't want the cable to be much shorter

Teeth are good for removing the outer isolation, just dont remove all, so be careful. Depending on the cable type you'll have one of three things in front of you:

2x mono cables put together like on the pic. The visible copper wire on both the two cables are to be twisted together, see next pic. The two wires with more isolation (if you bit it off, try again) are left and right channel (red and yellow color in the pic)

Stereo cable, has one copper wire (well technically a lot of small ones, twist them together :P ) and the two left/right isolated ones. Same as the one above, just not split into two wires (each for one channel).

Painted wires. IEM's, plugs and some headphones with thin cables (like on portapro) use paint instead of rubber as isolation. If you're reading this because you're a first timer at this, give up, as getting the paint off is ~impossible. Only way to do it is really to

twist the wire around the soldering iron and burn the paint off, not a good solution in the least.

Remove 3-4mm (1/8") of isolation from the left/right channel wires and twist it so its one wire and not a bunch of copper threads sticking out. Twist the non-isolated copper wire (which is ground, btw) into one wire if you have a split cable like mentioned above, and make sure this one doesn't have copper wires sticking out too.

Jackplug. Many versions are available at Radio Shack or similar stores. But better try to get some quality brand plugs, like Neutrik, Switchcraft, or Canare. They're not expensive and are very well manufactured.

Inside of jack. Looks pretty much the same no matter what plug you have. Has a big flange for the ground cable and two smaller for left/right channels.

IMPORTANT!
You don't want to finish this and realize you forgot to thread the end cap(s) on the cable before soldering, so thread them on in the right order. Depending on the jack, you will have one or more caps/protectors to thread on. The small plastic thing visible on the pic is because the casing of this jack is metal, and it's there to prevent short circuiting for that reason. I didn't find a stretch reliever, most plugs have one, either stuck to the end cap or as a separate part. It's there to prevent you from pulling the plug apart

The two small flanges have holes. Thread the exposed copper wires of the left/right channel into these holes (one channel into one hole, obviously). Try to do like on the pic, thread it from inside and out so that the copper wire sticks out, not in, to prevent it touching anything it shouldn't.
Depending on the plug you use, tracking what's left and right to the corresponding flanges shouldn't be a problem.

NOTE: Knowing the difference between left and right channel isn't always easy (especially if you don't have a split cable), if you don't know which of the audio cables are left and right, I've made a mp3 file for you to find out. Plug in your in progress plug, put on the headphones and check you got the right left/right "setup". The first 5 seconds is both channels, the next 5 is left channel, and the last 5 seconds are the right channel.

Solder the wires stuck. Just hold the soldering iron on the flange and touch it with the solder wire, not for too long just so a small drop melts and forms a stable solder point so the audio wire is steadily attached to the flange.

Should look something like this (actually should look prettier, I just did it in a hurry to get the pics :P)

After doing left and right channel, do the ground cable (the one without isolation) on the big flange. Same procedure, remember to pull the wire through the hole so far that the main isolation is close to the hole on the inside of the hole, that way there's no loose ground wire touching any of the left/right channel flanges. Solder stuck.

Cut off copper wire that sticks out on the left/right/ground flanges. So the solder points are clean.

The ground flange often has a secondary function as a small stretch protector, like this one. Use pliers to squeeze the main cable stuck to the ground flange, but don't overdue it

Depending on the plug as mentioned before, find the "thingies" you threaded on the cable earlier and screw the end cap on. Done, you have now changed the jack and it should work fine! If not, there might be a short circuit somewhere, open the plug and see if anything touches where it shouldn't. Neither the left, right or ground wire should be connected to one another in any way, like with a stray copper wire or a bent flange.